56 lines
3.0 KiB
TeX
56 lines
3.0 KiB
TeX
\chapter{Conclusion}%
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\label{chapter:conclusion}
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In the context of this thesis, two decoding algorithms were considered:
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proximal decoding and \ac{LP} decoding using \ac{ADMM}.
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The two algorithms were first analyzed individually, before comparing them
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based on simulation results as well as their theoretical structure.
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For proximal decoding, the effect of each parameter on the behavior of the
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decoder was examined, leading to an approach to choosing the value of each
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of the parameters.
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The convergence properties of the algorithm were investigated in the context
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of the relatively high decoding failure rate, to derive an approach to correct
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possible wrong componets of the estimate.
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Based on this approach, an improvement over proximal decoding was suggested,
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leading to a decoding gain of up to $\SI{1}{dB}$, depending on the code and
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the parameters considered.
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For \ac{LP} decoding using \ac{ADMM}, the circumstances brought about via the
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relaxation while formulating the \ac{LP} decoding problem were first explored.
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The decomposable nature arising from the relocation of the constraints into
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the objective function itself was recognized as the major driver in enabling
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the efficent implementation of the decoding algorithm.
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Based on simulation results, general guidelines for choosing each parameter
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were again derived.
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The decoding performance, in form of the \ac{FER}, of the algorithm was
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analyzed, observing that \ac{LP} decoding using \ac{ADMM} nearly reaches that
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of \ac{BP}, staying within approximately $\SI{0.5}{dB}$ depending on the code
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in question.
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Finally, strong parallells were discovered with regard to the theoretical
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structure of the two algorithms, both in the constitution of their respective
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objective functions as in the iterative approaches used to minimize them.
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One difference noted was the approximate nature of the minimization in the
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case of proximal decoding, leading to the constraints never being truly
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satisfied.
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In conjunction with the alternating minimization with respect to the same
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variable leading to oscillatory behavior, this was identified as the
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root cause of its comparatively worse decoding performance.
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Furthermore, both algorithms were expressed as message passing algorithms,
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justifying their similar computational performance.
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While the modified proximal decoding algorithm presented in section
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\ref{sec:prox:Improved Implementation} shows some promising results, further
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investigation is required to determine how different choices of parameters
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affect the decoding performance.
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Additionally, a more mathematically rigorous foundation for determining the
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potentially wrong components of the estimate is desirable.
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Another area benefiting from future work is the expantion of the \ac{ADMM}
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based \ac{LP} decoder into a decoder approximating \ac{ML} performance,
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using \textit{adaptive \ac{LP} decoding}.
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With this method, the successive addition of redundant parity checks is used
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to mitigate the decoder becoming stuck in erroneous solutions introduced due
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the relaxation of the constraints of the \ac{LP} decoding problem \cite{alp}.
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